Two-Dimensional Dirac Fermions Protected by Space-Time Inversion Symmetry in Black Phosphorus

We report the realization of novel symmetry-protected Dirac fermions in a surface-doped two-dimensional (2D) semiconductor, black phosphorus. The widely tunable band gap of black phosphorus by the surface Stark effect is employed to achieve a surprisingly large band inversion up to $\sim 0.6\mathrm{eV}$. High-resolution angle-resolved photoemission spectra directly reveal the pair creation of Dirac points and their movement along the axis of the glide-mirror symmetry. Unlike graphene, the Dirac point of black phosphorus is stable, as protected by space-time inversion symmetry, even in the presence of spin-orbit coupling. Our results establish black phosphorus in the inverted regime as a simple model system of 2D symmetry-protected (topological) Dirac semimetals, offering an unprecedented opportunity for the discovery of 2D Weyl semimetals.

Figure 1

Experimental band structure of BP near the $\mathrm{\Gamma }$ point along $k_y$ (a) before and (b) after the deposition of alkali-metal atoms at 1 ML. The dark yellow line in (a) shows an unoccupied CB of BP, obtained from tight-binding calculations. Blue and red dotted lines denote the maximum energy of ${\mathrm{\Gamma }}_2^{+}$ states and the minimum energy of ${\mathrm{\Gamma }}_4^{-}$ states, respectively, which are inverted between (a) and (b), as indicated by arrows. In (b), the spectral feature of the CB is well defined as compared to that of the VB owing to their real-space distribution [26, 27] and matrix-element effect. There are other shallow bands of BP near $E_F$, as predicted in first-principles band calculations [23]. (c) Constant-energy map at $E_D$ taken from BP at the same doping level as in (b). (d) Schematic illustration for the band structure of BP under vertical electric field. Solid squares show the surface Brillouin zone of BP with high-symmetry points marked by black dots and Dirac points marked by purple dots.

Figure 2

(a) 3D representation of ARPES data for the dopant density of 1 ML, where the Fermi surface and band dispersions across the ${\mathrm{\Lambda }}_x$ and ${\mathrm{\Lambda }}_y$ points are shown together. Dashed lines overlaid indicate ${\mathrm{\Gamma }}_2^{+}$ and ${\mathrm{\Gamma }}_4^{-}$ bands. High-resolution ARPES data for the band crossing of ${\mathrm{\Gamma }}_2^{+}$ and ${\mathrm{\Gamma }}_4^{-}$ states, taken at the ${\mathrm{\Lambda }}_y$ point (b) along $k_y$ and (c) along $k_x$, and (d) at the ${\mathrm{\Lambda }}_x$ point along $k_y$. (e) Normalized energy distribution curves of ARPES data shown in (d) with $0.022{\AA }^{-1}$ steps across the ${\mathrm{\Lambda }}_x$ point. Red and blue circles indicate the peak positions of ${\mathrm{\Gamma }}_2^{+}$ and ${\mathrm{\Gamma }}_4^{-}$ bands, respectively.

Figure 3

Theoretical band structure calculated from a tight-binding Hamiltonian of bilayer BP under vertical electric field ($U=1\mathrm{eV}$) in the (a) presence and (b) absence of $M_x$ symmetry. Red and blue lines denote ${\mathrm{\Gamma }}_2^{+}$ and ${\mathrm{\Gamma }}_4^{-}$ bands, respectively. Inset in (a) shows a tiny $k$ splitting of Weyl points with respect to the ${\mathrm{\Lambda }}_y$ point in the presence of spin-orbit interaction [23]. Inset in (b) shows the glide-mirror plane of $M_x$ symmetry in the unit structure of BP. (c) Schematic illustration showing that the stable Dirac point of BP splits into a pair of Weyl points in the presence of spin-orbit coupling, which is different from that of graphene [7].

Figure 4

The pair creation and moving of Dirac points. The $k$ splitting of Dirac points is plotted as a function of the dopant density. A fit with the power-law function (black line overlaid) yields the exponent value of 0.48, confirming the monotonic energy shift of ${\mathrm{\Gamma }}_2^{+}$ and ${\mathrm{\Gamma }}_4^{-}$ bands. Inset shows the merging of two Dirac cones, resulting in the linear-quadratic band dispersion at the critical point.